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Math 241, Quiz 8. 10/22/12. Name: • Read problems carefully. Show all work. No notes, calculator, or text. • There are 15 points total. §14.8, #33 (15 points): Use Lagrange Multipliers to find the maximum volume of a rectangular box that is inscribed in a sphere of radius r. Solution: The problem is to maximize V = (2x)(2y)(2z) = 8xyz subject to the constraint g = x2 + y 2 + z 2 = r2 . We compute ∇V = h8yz, 8xz, 8xyi, ∇g = h2x, 2y, 2zi. With λ 6= 0 and x, y, z > 0 (as spatial dimensions), we solve the system 4yz = λx, 4xz = λy, 4xy = λz, x2 + y 2 + z 2 = r2 . Since x, y, z 6= 0, we solve for λ in the first three equations to obtain 4xz 4xy 4yz = = = λ. x y z We take equations (1) and (2) together to get 4y 2 z = 4x2 z. Since z 6= 0, we find that y 2 = x2 . Similarly, using equations (1) and (3) gives x2 = z 2 ; we conclude that x2 = y 2 = z 2 . We substitute 2 2 in the fourth equation in the original system to get 3x √ √ =3 r . Taking √ the positive square root gives 3 x = y = z = r/ 3; hence, the volume is V = 8(r/ 3) = 8r /3 3.